Two types of invariant Subspaces in the polydisc

TL;DR

This paper introduces two new classifications of invariant subspaces in the Hardy space over the polydisc, providing characterizations and studying their unitary equivalence to better understand their complex structure.

Contribution

It defines two types of invariant subspaces in $H^2( ext{D}^n)$ and characterizes them using the Beurling-Lax-Halmos Theorem, advancing the understanding of their structure.

Findings

Characterization of the two types of invariant subspaces.

Analysis of unitary equivalence among these subspaces.

Enhanced understanding of the structure of invariant subspaces in the polydisc.

Abstract

It is known that the structure of invariant subspaces of the Hardy space $H^2(\mathbb D^n)$ on the polydisc $\mathbb{D}^n$ is very complicated; hence, we need good examples help us to understand the structure of invariant subspaces of $H^2(\mathbb D^n)$. In this paper, we define two types of invariant subspaces of $H^2(\mathbb D^n)$. Then, we give a characterization of these types invariant subspaces in view of the Beurling-Lax-Halmos Theorem. Unitary equivalence is also studied in this paper.